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Properties

Label 14.0.39249600651...3827.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,547^{13}$
Root discriminant $348.67$
Ramified prime $547$
Class number $32259$ (GRH)
Class group $[32259]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4124434193, -1079662218, 716216243, 755961898, 245440053, 5224933, -6262919, -914451, 92865, -4668, -1238, -162, 20, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 20*x^12 - 162*x^11 - 1238*x^10 - 4668*x^9 + 92865*x^8 - 914451*x^7 - 6262919*x^6 + 5224933*x^5 + 245440053*x^4 + 755961898*x^3 + 716216243*x^2 - 1079662218*x + 4124434193)
 
gp: K = bnfinit(x^14 - x^13 + 20*x^12 - 162*x^11 - 1238*x^10 - 4668*x^9 + 92865*x^8 - 914451*x^7 - 6262919*x^6 + 5224933*x^5 + 245440053*x^4 + 755961898*x^3 + 716216243*x^2 - 1079662218*x + 4124434193, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 20 x^{12} - 162 x^{11} - 1238 x^{10} - 4668 x^{9} + 92865 x^{8} - 914451 x^{7} - 6262919 x^{6} + 5224933 x^{5} + 245440053 x^{4} + 755961898 x^{3} + 716216243 x^{2} - 1079662218 x + 4124434193 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-392496006513281583932595840500243827=-\,547^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $348.67$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $547$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(547\)
Dirichlet character group:    $\lbrace$$\chi_{547}(544,·)$, $\chi_{547}(1,·)$, $\chi_{547}(546,·)$, $\chi_{547}(3,·)$, $\chi_{547}(520,·)$, $\chi_{547}(9,·)$, $\chi_{547}(365,·)$, $\chi_{547}(304,·)$, $\chi_{547}(81,·)$, $\chi_{547}(466,·)$, $\chi_{547}(243,·)$, $\chi_{547}(182,·)$, $\chi_{547}(538,·)$, $\chi_{547}(27,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{4}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{3} + \frac{3}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{11} a^{9} - \frac{4}{11} a^{7} + \frac{3}{11} a^{6} + \frac{3}{11} a^{5} + \frac{3}{11} a^{4} - \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{2}{11} a$, $\frac{1}{11} a^{10} + \frac{4}{11} a^{7} + \frac{1}{11} a^{6} - \frac{2}{11} a^{5} - \frac{5}{11} a^{4} - \frac{3}{11} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{121} a^{12} + \frac{3}{121} a^{11} - \frac{5}{121} a^{9} - \frac{4}{121} a^{8} + \frac{52}{121} a^{7} + \frac{20}{121} a^{6} + \frac{45}{121} a^{5} - \frac{12}{121} a^{4} + \frac{7}{121} a^{3} + \frac{6}{121} a^{2} - \frac{25}{121} a$, $\frac{1}{18772216479128563719492890527098392895422363461692923} a^{13} + \frac{31430287000326282984261188598834650619122301556833}{18772216479128563719492890527098392895422363461692923} a^{12} + \frac{42975666076990082215595128621373307741550338318409}{18772216479128563719492890527098392895422363461692923} a^{11} - \frac{4062641464018873765945493376588079903142302073713}{18772216479128563719492890527098392895422363461692923} a^{10} - \frac{637923245077914216543779502246039247487926750249917}{18772216479128563719492890527098392895422363461692923} a^{9} + \frac{526901877059188216096715231148047817534632232332307}{18772216479128563719492890527098392895422363461692923} a^{8} + \frac{4385737867685768096229169640163303287695814303404296}{18772216479128563719492890527098392895422363461692923} a^{7} + \frac{7628982801495354260464298517615522980734537357005046}{18772216479128563719492890527098392895422363461692923} a^{6} + \frac{15268822117387055161977488289786774184666592224415}{95290438980348039185243099122326867489453621632959} a^{5} - \frac{4410961279591839468613569037836960185736180706611549}{18772216479128563719492890527098392895422363461692923} a^{4} + \frac{2981023133967658229224142608506254685806394814098699}{18772216479128563719492890527098392895422363461692923} a^{3} - \frac{2665276641703020058819570590857728134465962441634229}{18772216479128563719492890527098392895422363461692923} a^{2} - \frac{2081094231199654303073606139270452134906595394970434}{18772216479128563719492890527098392895422363461692923} a - \frac{1532849511482485849517684979760482787122666108942}{3300899679818632621679776776349286600214939944029}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{32259}$, which has order $32259$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 69192405.98699488 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-547}) \), 7.7.26786992617986329.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
547Data not computed